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Theorem tpid3g 3577
Description: Closed theorem form of tpid3 3578. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
tpid3g (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})

Proof of Theorem tpid3g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elisset 2647 . 2 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
2 3mix3 1117 . . . . . . 7 (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
32a1i 9 . . . . . 6 (𝐴𝐵 → (𝑥 = 𝐴 → (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)))
4 abid 2083 . . . . . 6 (𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)} ↔ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴))
53, 4syl6ibr 161 . . . . 5 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}))
6 dftp2 3511 . . . . . 6 {𝐶, 𝐷, 𝐴} = {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)}
76eleq2i 2161 . . . . 5 (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝑥 ∈ {𝑥 ∣ (𝑥 = 𝐶𝑥 = 𝐷𝑥 = 𝐴)})
85, 7syl6ibr 161 . . . 4 (𝐴𝐵 → (𝑥 = 𝐴𝑥 ∈ {𝐶, 𝐷, 𝐴}))
9 eleq1 2157 . . . 4 (𝑥 = 𝐴 → (𝑥 ∈ {𝐶, 𝐷, 𝐴} ↔ 𝐴 ∈ {𝐶, 𝐷, 𝐴}))
108, 9mpbidi 150 . . 3 (𝐴𝐵 → (𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
1110exlimdv 1754 . 2 (𝐴𝐵 → (∃𝑥 𝑥 = 𝐴𝐴 ∈ {𝐶, 𝐷, 𝐴}))
121, 11mpd 13 1 (𝐴𝐵𝐴 ∈ {𝐶, 𝐷, 𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 926   = wceq 1296  wex 1433  wcel 1445  {cab 2081  {ctp 3468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3or 928  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-un 3017  df-sn 3472  df-pr 3473  df-tp 3474
This theorem is referenced by:  rngmulrg  11777  srngmulrd  11784  lmodscad  11795  ipsmulrd  11803  ipsipd  11806  topgrptsetd  11813
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